How do you factor the expression #5+34x-7x^2#?

Question

Eva Abramson · Accepted Answer

#-(7x-1)# and #(x-5)# We begin with #5+34x-7x^2#. I am not used to seeing a problem written this way, so I am going to rewrite it into standard form (#ax^2+bx+c#). If we do that, we end with #-7x^2+34x+5#. From here, I'm going to do one last thing: I'm going to factor out that negative in front of the #7x^2#. I'm doing that because I don't want to have to deal with it, and because I am not actually changing the expression, just rewriting it. NOW we have something I'm ready to factor: #-(7x^2-34x-5)#. Let's get started with factoring. Now, if #-(7x^2-34x-5)# is in standard form, then #7# would be #a#, #-34# would be #b#, and #-5# would be #c#. Keep that in mind when i discuss factoring. Anyways, the first thing I always do is find two numbers that can be multiplied to equal the #a*b# and can be added to equal #c#. So, what does #a*b# equal? For us, that's #7*-5#, which is #-35#. And in our case, #c# is #-34#. Okay, we know what we're looking for: two numbers that add to #-34# and multiply to #-35#, and I think #1*-35# fits those constraints! From here, we just write the first and last values with space left for the numbers we just found, like this: #(7x^2+ ____x)+(____x+-5)#. We can insert the #-35# and the #1# wherever we want. I'm going to put the #1# with the #7x^2# and the #-35# with the #-5#, but you can do it any way you want. We now have #(7x^2+ 1x)+(-35x+-5)#. I see that we can factor out an #x# from the first parentheses and a #-5# from the second. that gives us #x(7x+1)+ -5(7x+1)#. We now take the factored values (the #x# and the #-5#) and write them alongside the other group (#7x+1#) to give us #(x-5)(7x+1)#. We are almost done, save for the fact that the whole problem looks like this #-((x+1)(7x+1))#. We multiply in the negative for one of the groups (I picked #7x+1#). That leaves us with #(-7x-1)(x+1)#. Now we are done.

Genesis Allen · Answer

undefined To factor the expression 5 + 34x - 7x^2, you can follow these steps:

1. Rearrange the terms in descending order of the powers of x: -7x^2 + 34x + 5.
2. Identify the common factor, if any. In this case, there is no common factor among the terms.
3. Use the quadratic formula to factor the quadratic expression. The quadratic formula is given by: $x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$.
4. Substitute the coefficients $a$, $b$, and $c$ from the quadratic expression into the quadratic formula.
5. Solve for the values of $x$ using the quadratic formula.
6. Rewrite the quadratic expression as the product of its factors using the solutions obtained from step 5.
7. Check the factored expression by multiplying the factors to ensure they produce the original expression.